<?xml version="1.0"?>
<div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>,  <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">$1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">$2 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">A</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> ;<br/>
<a NAME="E1:188"/><span class="lab"><font color="Green" title="E130">A1</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(3,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E2:188"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,2</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>,  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default">55 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">84 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>,  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default">36 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">55 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>] <span class="kw">means </span>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">H<sub>2</sub></font>($2 <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ,$2 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ),<font color="Maroon">H<sub>3</sub></font>($2 <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ,$2 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> )</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f</font> =  <a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>);<br/>
<a NAME="E4:188"/><span class="lab"><font color="Green" title="E131">A2</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</font>;<br/>
<a NAME="E5:188"/>
<span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,2</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#D10" target="_self" onmouseover="rs('number08/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E6:188"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A3</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:188"><span class="lab"><font color="Green" title="E130">A1</font></span></a></span>;<br/>
<a NAME="E7:188"/><span class="lab"><font color="Green" title="E133">A4</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="188_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:188_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:188_1"/><span class="lab"><font color="Green" title="E134">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> ;<br/>
<a NAME="E4:188_1"/><span class="lab"><font color="Green" title="E135">A6</font></span>: 
<span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> ),<font color="Maroon">H<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> )</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#D10" target="_self" onmouseover="rs('number08/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E5:188_1"/><span class="lab"><font color="Green" title="E136">A7</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T89" target="_self" onmouseover="rs('number08/T89')" onmouseout="rh()">Th89</a></span>;<br/>
<a NAME="E6:188_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> ),<font color="Maroon">H<sub>3</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> )) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:188_1"><span class="lab"><font color="Green" title="E134">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:188_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:188_1"><span class="lab"><font color="Green" title="E135">A6</font></span></a>, <a class="txt" href="#E5:188_1"><span class="lab"><font color="Green" title="E136">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:188"/><span class="lab"><font color="Green" title="E134">A8</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E6:188"><span class="lab"><font color="Green" title="E132">A3</font></span></a>, <a class="txt" href="#E7:188"><span class="lab"><font color="Green" title="E133">A4</font></span></a>);<br/></span>
<a NAME="E9:188"/><span class="lab"><font color="Green" title="E135">A9</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="188_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:188_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font></span><br/>

<a NAME="E2:188_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:188_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:188"><span class="lab"><font color="Green" title="E134">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:188"/>
 <a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (3,2,55,84,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,36,55,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T92" target="_self" onmouseover="rs('number08/T92')" onmouseout="rh()">Th92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:188"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:188"><span class="lab"><font color="Green" title="E131">A2</font></span></a>, <a class="txt" href="#E9:188"><span class="lab"><font color="Green" title="E135">A9</font></span></a>, <a class="ref" href="card_1.html#T59" onmouseover="rs('card_1/T59')" onmouseout="rh()">CARD_1:59</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
